On the Monotonicity of the Generalized Marcum and Nuttall Q-Functions

Monotonicity criteria are established for the generalized Marcum Q-function, \emph{Q}_{M}, the standard Nuttall Q-function, \emph{Q}_{M,N}, and the normalized Nuttall Q-function, $\mathcal{Q}_{M,N}$, with respect to their real order indices M,N. Besides, closed-form expressions are derived for the computation of the standard and normalized Nuttall Q-functions for the case when M,N are odd multiples of 0.5 and $M\geq N$. By exploiting these results, novel upper and lower bounds for \emph{Q}_{M,N} and $\mathcal{Q}_{M,N}$ are proposed. Furthermore, specific tight upper and lower bounds for \emph{Q}_{M}, previously reported in the literature, are extended for real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the information-theoretic analysis of multiple-input multiple-output systems and in the description of stochastic processes in probability theory, among others.Comment: Published in IEEE Transactions on Information Theory, August 2009. Only slight formatting modification

Bounds for the symmetric difference of generalized Marcum Q-functions

Recently, an approximation for large values of a and b for the symmetric difference of Marcum Q-functions Q�(a; b) was given in [1] in the case of integer order, i.e. when � = n 2 N. Motivated by this result, in this note we study the symmetric difference of Marcum Q-functions Q�(a; b) of real order �1 for the parameters a > b > 0. Our aim is to use some of the lower and upper bounds of the Marcum Q-function that appear in the literature to obtain some tight bounds for the symmetric difference. Another approach, presented in this note, is to investigate the difference via closed forms of the Marcum Q-function

Asymptotically Exact Approximations for the Symmetric Difference of Generalized Marcum-Q Functions

(c) 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. DOI: 10.1109/TVT.2014.2337263In this paper, we derive two simple and asymptotically exact approximations for the function defined as ΔQm(a, b) =Δ Qm(a, b) - Qm(b, a). The generalized Marcum Q-function Qm(a, b) appears in many scenarios in communications in this particular form and is referred to as the symmetric difference of generalized Marcum Q-functions or the difference of generalized Marcum Q-functions with reversed arguments. We show that the symmetric difference of Marcum Q-functions can be expressed in terms of a single Gaussian Q-function for large and even moderate values of the arguments a and b. A second approximation for ΔQm(a, b) is also given in terms of the exponential function. We illustrate the applicability of these new approximations in different scenarios: 1) statistical characterization of Hoyt fading; 2) performance analysis of communication systems; 3) level crossing statistics of a sampled Rayleigh envelope; and 4) asymptotic approximation of the Rice Ie-function.Universidad de Málaga. Campus de Excelencia Internacional. Andalucía Tech

Analytic Expressions and Bounds for Special Functions and Applications in Communication Theory

This paper is devoted to the derivation of novel analytic expressions and bounds for a family of special functions that are useful in wireless communication theory. These functions are the well-known Nuttall Q-function, incomplete Toronto function, Rice Ie-function, and incomplete Lipschitz-Hankel integrals. Capitalizing on the offered results, useful identities are additionally derived between the above functions and Humbert, Φ1, function as well as for specific cases of the Kampé de Fériet function. These functions can be considered as useful mathematical tools that can be employed in applications relating to the analytic performance evaluation of modern wireless communication systems, such as cognitive radio, cooperative, and free-space optical communications as well as radar, diversity, and multiantenna systems. As an example, new closed-form expressions are derived for the outage probability over nonlinear generalized fading channels, namely, α-η-μ, α-λ-μ, and α-κ-μ as well as for specific cases of the η-μ and λ-μ fading channels. Furthermore, simple expressions are presented for the channel capacity for the truncated channel inversion with fixed rate and corresponding optimum cutoff signal-to-noise ratio for single-antenna and multiantenna communication systems over Rician fading channels. The accuracy and validity of the derived expressions is justified through extensive comparisons with respective numerical results

Algorithms for improving the efficiency of CEV, CIR and JDCEV option pricing models

The non-central chi-square distribution function has extensive use in the field of Mathematical Finance. To a great extent, this is due to its involvement in the constant elasticity of variance (hereafter, CEV) option pricing model of Cox (1975), in the term structure of interest rates model of Cox et al. (1985a) (hereafter, CIR), and the jump to default extended CEV (hereafter, JDCEV) framework of Carr and Linetsky (2006). Efficient computation methods are required to rapidly price complex contracts and calibrate financial models. The processes with several parameters, like the CEV or JDCEV models that we will address are examples of where this is important, since in this case the pricing problem (for many strikes) is used inside an optimization method. With this work we intend to test recent developments concerning the efficient computation of the non-central chi-square distribution function in the context of these option pricing models. We will give particular emphasis to the recent developments presented in the work of Gil et al. (2012), Gil et al. (2013), Dias and Nunes (2014), and Gil et al. (2015). For each option pricing model, we will define reference data-sets compatible with the most common combination of values used in pricing practice, following a framework that is similar to the one presented in Larguinho et al. (2013). We will conclude by offering novel analytical solutions for the JDCEV delta hedge ratios for the recovery parts of the put.A distribuição de probabilidade chi-quadrado não-central tem sido alvo de vasta utilização no domínio da Matematica Financeira, em grande parte devido à sua utilização no modelo constant elasticity of variance (doravante, CEV) de Cox (1975), no term structure of interest rates model de Cox et al. (1985a) e no modelo jump to default extended CEV (doravante, JDCEV) de Carr and Linetsky (2006). Metodos de cálculo eficientes e rápidos são de especial relevancia na calibração de modelos para a determinação do preço de contratos financeiros complexos. Os modelos CEV, CIR e JDCEV sao exemplos de modelos com diversos parametros que, quando usados em contexto de determinação do preço de opções com vários preçoss de exercício, mostram como esta optimização e fundamental. Com este trabalho pretendemos testar os mais recentes desenvolvimentos no calculo eficiente da distribuição de probabilidade nao-central chi-quadrado, no contexto dos modelos de cálculo de preço de opções mencionados anteriormente. Daremos enfase aos recentes desenvolvimentos apresentados nos trabalhos de Gil et al. (2012), Gil et al. (2013), Dias and Nunes (2014) e de Gil et al. (2015). Para cada um dos modelos, definiremos um conjunto de parametros de referencia compativel com as combinações mais usadas na prática, seguindo uma metodologia similiar a usada em Larguinho et al. (2013). Concluímos com a derivação de novas soluções analíticas para os racios de delta hedging no modelo JDCEV